nLab superoperator

Idea

In quantum physics and in particular in the context of quantum computing a linear map between spaces of linear operators (hence an “operator on operators”) is sometimes called a superoperator.

Notably density matrices representing quantum states are given by linear operators and a quantum operation is a superoperator that preserves the defining properties of density matrices. (Beware that sometimes different or no distinctions between “quantum operation” and “superoperator” are made.)

Details

Specifically, in a symmetric monoidal closed category $(\mathcal{C}, \otimes, \mathbb{1})$ with internal hom denoted $(\text{-})\multimap (\text{-})$, a superoperator is a morphism of a form like

(1)$$(\mathscr{H} \multimap \mathscr{H}) \longrightarrow (\mathscr{K} \multimap \mathscr{K}) \,.$$

If $(\mathcal{C}, \otimes, \mathbb{1})$ is also compact closed then (1) is isomorphic to a morphism of the form

(2)$$\mathscr{H} \otimes \mathscr{H}^\ast \longrightarrow \mathscr{K} \otimes \mathscr{K}^\ast$$

(whence one also speaks of operators on spaces of (density) matrices)

which in turn is adjunct, in particular, to a morphism of the form

(3)$$\mathscr{K}^\ast \otimes \mathscr{H} \longrightarrow \mathscr{K}^\ast \otimes \mathscr{H} \mathrlap{\,.}$$

Sometimes (3) is the more transparent incarnation of superoperators:

For example, if $(\mathcal{C}, \otimes, \mathbb{1})$ is even a dagger-compact category, then a superoperator in the form (1) or (2) is a completely positive map (and hence restricts to a “quantum operation” on density matrices) iff its adjunct (3) is a positive operator [Selinger 2005 Cor. 4.13].

Related entries

• The Geometry of Interaction-construction is a genral abstract construction of a category of superoperators. See there for more.

References

Traditional discussion:

• Karol Życzkowski, Ingemar Bengtsson, Section 4 of: On Duality between Quantum Maps and Quantum States, Open Systems & Information Dynamics 11 01 (2004) 3-42 [doi:10.1023/B:OPSY.0000024753.05661.c2]

• Greg Kuperberg, Section 1.4 of: A concise introduction to quantum probability, quantum mechanics, and quantum computation (2005) [pdf, pdf]

See also:

• Wikipedia, Superoperator

In terms of string diagrams in dagger-compact categories (cf. quantum information theory via dagger-compact categories):

• Peter Selinger, Dagger-compact closed categories and completely positive maps, Electronic Notes in Theoretical Computer Science, 170 (2007) 139-163 [doi:10.1016/j.entcs.2006.12.018, pdf]

Formalizing superoperators as arrows (in computer science):

• J. K. Vizzotto, Thorsten Altenkirch, A. Sabry, Structuring quantum effects: superoperators as arrows, Mathematical Structures in Computer Science 16 3 (2006) 453-468 [arXiv:quant-ph/0501151, doi:10.1017/S0960129506005287]